Will day one stay the record all week?

Question

Five independent daily values arrive in order, each from the same continuous distribution (no ties).

What is the probability that no later day ever beats day 1, that is, day 1's value stays the record for all five days?

Accepted Answer

Day 1 stays the record through all five days exactly when day 1's value is the largest of the five. All five values are exchangeable, so each is equally likely to be the biggest, giving P(	ext{day 1 is the maximum}) = \frac{1}{5} = 0.2. You can also multiply the "no new record" chances day by day: \left(1-	frac12ight)\left(1-	frac13ight)\left(1-	frac14ight)\left(1-	frac15ight) = 	frac12\cdot	frac23\cdot	frac34\cdot	frac45 = 	frac15, since the product telescopes. Takeaway. The opener survives as the record over n draws with probability 1/n; the day-by-day telescoping product …

Will day one stay the record all week?

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